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It has one.
An empty cone (like an ice-cream cone) doesn't have any flat surfaces. It has two curved surfaces, one outside and one inside. A solid cone has a flat base, and one other curved surface.
a cone has 2 surfaces Surfaces: 2 (the base and the slanted surface) Edges: 1 (the joining of base with the slanted surface) Corners: 1 (the apex)
None. There is one curved suface but no plane surface.
If it is a right circular cone, it has an infinite number of planes of symmetry. If it is an oblique circular cone, it has one plane of symmetry.